∫ calculus

1. نبدأ بحل التكامل التالي: $$\int \sqrt{9 - 4x^2} \, dx$$. 2. نلاحظ أن التعبير تحت الجذر هو من الشكل $$a^2 - (bx)^2$$ حيث $$a=3$$ و $$b=2$$.

1. **State the problem:** Find the limit as $x$ approaches 2 of the function $$\frac{x^2 - 3x + 2}{x - 2}.$$\n\n2. **Recall the formula and rules:** The limit of a rational functio

1. The problem involves evaluating the triple integral with limits for $x$ from 2 to -2, for $y$ from $\sqrt{4-x}$ to $\sqrt{4-x^2}$, and for $z$ from 3 to 0, integrating the funct

1. The problem involves evaluating the triple integral of the function $f(x,y,z) = 2z$ over the region defined by the limits for $z$, $y$, and $x$ given as $z$ from 2 to -2, $y$ fr

1. **Evaluate the indefinite integral** \(\int (x^3 + 5) \, dx\). 2. The integral of a sum is the sum of the integrals:

1. समस्या: एक गोलाकार गुब्बारे की त्रिज्या $r$ समय $t$ के साथ स्थिर गति से बदल रही है। आरंभ में $r=3$ ईकाई और 3 सेकंड बाद $r=6$ ईकाई है। $t$ सेकंड बाद त्रिज्या ज्ञात करें। 2. सूत्र

1. The problem is to find the integral $$\int e^x \sin x \, dx$$. 2. We use integration by parts or recognize this as a standard integral involving exponential and trigonometric fu

1. The problem: You asked for a list of first principles formulas and to highlight the important ones. 2. First principles in calculus refer to the definition of derivatives using

1. **Statement of the problem:** We are given that the curve \((C_f)\) of the function \(f(x) = (x+1)e^{-x} - x\) intersects the x-axis at two points with abscissas \(x_1\) and \(x

1. समस्या 15: बिंदु (0,0) से गुजरने वाली रेखा का समीकरण ज्ञात करें, जिसका अवकल व्युत्पन्न $y' = e^x \sin x$ है। 2. सूत्र: यदि $\frac{dy}{dx} = f(x)$ हो, तो $y = \int f(x) dx + C$.

1. نبدأ بحساب معادلة المماس (T) للمنحني (C_f) عند النقطة ذات الفاصلة 0. - الدالة المعطاة: $$f(x) = (x+1)e^{-x} - x$$

1. समस्या: बिंदु (0, 0) से गुजरने वाले वक्र का समीकरण ज्ञात करें जिसका अवकल समीकरण $y' = e^x \sin x$ है। 2. अवकल समीकरण का अर्थ है $\frac{dy}{dx} = e^x \sin x$। इसे हल करने के लिए

1. Stating the problem: Find $\frac{dy}{dx}$ for the function given by $y^3 = 4x$. 2. Formula and rules: We will use implicit differentiation since $y$ is given implicitly by the e

1. نعيد صياغة السؤال 4: بيّن أن $f(\alpha) = -\alpha - 1 - \frac{1}{\alpha}$ ثم عين حصراً قيمة $f(\alpha)$.\n\n2. نذكر أن $\alpha$ هو الحل الوحيد للمعادلة $g(x) = 0$ في المجال $]-\

1. **بيان المسألة:** لدينا دالتان معرفتان على مجموعات معينة:

1. **بيان المسألة:** لدينا الدالة $g(x) = e^{-x} + \frac{1}{x}$ معرفة على $\mathbb{R} \setminus \{0\}$. المطلوب: - حساب نهاية $g(x)$ عند أطراف مجال التعريف.

1. نبدأ بتحديد المشكلة: نريد إيجاد مشتقة الدالة $$f(x) = (\ln x)^2$$. 2. نستخدم قاعدة السلسلة للمشتقة، حيث إذا كانت الدالة على شكل $$g(h(x))$$ فإن مشتقتها هي $$g'(h(x)) \cdot h'(x)

1. لنبدأ بتحديد المشكلة: نريد إيجاد مشتق الدالة $\ln(x)$. 2. القاعدة الأساسية لمشتقة اللوغاريتم الطبيعي هي:

1. **State the problem:** Solve the differential equation $$(x^3+x^2+x+1)\frac{dy}{dx} = 2x^2 + x$$ with the initial condition $x=0$, $y=1$. 2. **Rewrite the equation:** Isolate $\

1. **State the problem:** Find the first-quadrant point on the curve defined by the equation $y^2 x = 18$ that is closest to the point $(2,0)$. 2. **Understand the problem:** We wa

1. Vamos calcular a área da região R para cada função dada, usando a integral definida quando necessário. (a) Para $f(x) = 4 - 2x$ no intervalo $[0,2]$: